So why don't they just send us a bill saying "you owe this", and be done with it?

Because they're sometimes wrong. Many years ago, my wife and I were in-person audited. At the end of the audit, it turned out the IRS owed us a few bucks. We've been letter-audited a number of times over the years, once with the IRS claiming we owed five figures. In that case, it turned out the IRS agent had misinterpreted the stock symbols; we owed nothing. The other times, the IRS wound up owing us a couple of dollars, for which they sent us a check, and on other occasions, it turned out we owed them a couple of dollars, which they waived on their own initiative because the amount was below their threshold for caring about.

Years ago, the British tax authority floated a variation on your idea: have the employers send the top-line amounts owed each employee directly to the tax authority, they'd figure the taxes owed, take that, and send to the employee what was left. Trust the government. Fortunately, the rest of the government listened to the hue and cry and scotched the idea.

Eric Hines